on a 425, yes, it should run whenever the key is on, regardless if the engine is running or not, and regardless of the position of the brake pedal, or any other switch. 425 is carbureted, and has a low pressure pump, which keeps the bowl on the carb full, but can't build enough pressure to force open the float and needle valve.
If you had a 445, which is fuel injected, then no, the pump only runs for a few seconds, then shuts off if the engine is not running. When you are cranking the engine, the pump runs, and when the engine is running it runs. This is a high pressure pump, as it needs to supply sufficent pressure to the fuel injectors.